ELASTICITIES AND LOGARITHMIC MODELS
1
This sequence defines elasticities and shows how one may fit nonlinear models with constant elasticities. First, the general definition of an elasticity.
Y
X
A
0 52O
Definition:
The elasticity of Y with respect to X is the proportional change in Y per proportional change in X.
XYdXdY
XdXYdY
elasticity
XY
2
Re-arranging the expression for the elasticity, we can obtain a graphical interpretation.
ELASTICITIES AND LOGARITHMIC MODELS
Y
X
A
0 52O
Definition:
The elasticity of Y with respect to X is the proportional change in Y per proportional change in X.
XYdXdY
XdXYdY
elasticity
XY
slope of the tangent at Aslope of OA
Y
X
A
3
The elasticity at any point on the curve is the ratio of the slope of the tangent at that point to the slope of the line joining the point to the origin.
0 52O
Definition:
The elasticity of Y with respect to X is the proportional change in Y per proportional change in X.
XYdXdY
XdXYdY
elasticity
ELASTICITIES AND LOGARITHMIC MODELS
XY
slope of the tangent at Aslope of OA
4
In this case it is clear that the tangent at A is flatter than the line OA and so the elasticity must be less than 1.
ELASTICITIES AND LOGARITHMIC MODELS
Y
X
A
0 52O
Definition:
The elasticity of Y with respect to X is the proportional change in Y per proportional change in X.
XYdXdY
XdXYdY
elasticity
XY
slope of the tangent at Aslope of OA
elasticity < 1
0 52
5
In this case the tangent at A is steeper than OA and the elasticity is greater than 1.
A
O
Y
X
elasticity > 1
ELASTICITIES AND LOGARITHMIC MODELS
Definition:
The elasticity of Y with respect to X is the proportional change in Y per proportional change in X.
XYdXdY
XdXYdY
elasticity
slope of the tangent at Aslope of OA
6
In general the elasticity will be different at different points on the function relating Y to X.
ELASTICITIES AND LOGARITHMIC MODELS
XY 21
21
2
21
2
)/(
/)(
X
XX
slope of the tangent at Aslope of OA
XYdXdY
elasticity
x
A
O
Y
X
7
In the example above, Y is a linear function of X.
ELASTICITIES AND LOGARITHMIC MODELS
x
A
O
Y
X
XY 21
21
2
21
2
)/(
/)(
X
XX
slope of the tangent at Aslope of OA
XYdXdY
elasticity
8
The tangent at any point is coincidental with the line itself, so in this case its slope is always 2. The elasticity depends on the slope of the line joining the point to the origin.
ELASTICITIES AND LOGARITHMIC MODELS
x
A
O
Y
X
XY 21
21
2
21
2
)/(
/)(
X
XX
slope of the tangent at Aslope of OA
XYdXdY
elasticity
9
OB is flatter than OA, so the elasticity is greater at B than at A. (This ties in with the mathematical expression: (1/ X) + 2 is smaller at B than at A, assuming that 1 is positive.)
x
A
O
B
Y
X
ELASTICITIES AND LOGARITHMIC MODELS
XY 21
21
2
21
2
)/(
/)(
X
XX
slope of the tangent at Aslope of OA
XYdXdY
elasticity
10
However, a function of the type shown above has the same elasticity for all values of X.
ELASTICITIES AND LOGARITHMIC MODELS
21
XY
11
For the numerator of the elasticity expression, we need the derivative of Y with respect to X.
ELASTICITIES AND LOGARITHMIC MODELS
21
XY
121
2 XdXdY
12
For the denominator, we need Y/X.
ELASTICITIES AND LOGARITHMIC MODELS
21
XY
121
2 XdXdY
11
1 2
2
X
XX
XY
13
Hence we obtain the expression for the elasticity. This simplifies to 2 and is therefore constant.
21
XY
121
2 XdXdY
11
1 2
2
X
XX
XY
elasticity 211
121
2
2
X
XXYdXdY
ELASTICITIES AND LOGARITHMIC MODELS
14
By way of illustration, the function will be plotted for a range of values of 2. We will start with a very low value, 0.25.
Y
X
ELASTICITIES AND LOGARITHMIC MODELS
21
XY 25.02
15
Y
X
ELASTICITIES AND LOGARITHMIC MODELS
21
XY 50.02
We will increase 2 in steps of 0.25 and see how the shape of the function changes.
16
Y
X
ELASTICITIES AND LOGARITHMIC MODELS
21
XY
2 = 0.75.
75.02
17
Y
X
ELASTICITIES AND LOGARITHMIC MODELS
21
XY
When 2 is equal to 1, the curve becomes a straight line through the origin.
00.12
18
Y
X
ELASTICITIES AND LOGARITHMIC MODELS
21
XY
2 = 1.25.
25.12
19
Y
X
ELASTICITIES AND LOGARITHMIC MODELS
21
XY
2 = 1.50.
50.12
20
Y
X
ELASTICITIES AND LOGARITHMIC MODELS
21
XY
2 = 1.75. Note that the curvature can be quite gentle over wide ranges of X.
75.12
21
Y
X
ELASTICITIES AND LOGARITHMIC MODELS
21
XY 75.12
This means that even if the true model is of the constant elasticity form, a linear model may be a good approximation over a limited range.
22
It is easy to fit a constant elasticity function using a sample of observations. You can linearize the model by taking the logarithms of both sides.
ELASTICITIES AND LOGARITHMIC MODELS
X
X
XY
loglog
loglog
loglog
21
1
1
2
2
21
XY
23
You thus obtain a linear relationship between Y' and X', as defined. All serious regression applications allow you to generate logarithmic variables from existing ones.
ELASTICITIES AND LOGARITHMIC MODELS
X
X
XY
loglog
loglog
loglog
21
1
1
2
2
'' 2'1 XY
1'1 log
log'
,log'
XX
YYwhere
21
XY
24
The coefficient of X' will be a direct estimate of the elasticity, 2.
ELASTICITIES AND LOGARITHMIC MODELS
X
X
XY
loglog
loglog
loglog
21
1
1
2
2
21
XY
'' 2'1 XY
1'1 log
log'
,log'
XX
YYwhere
25
The constant term will be an estimate of log 1. To obtain an estimate of 1, you calculate exp(b1'), where b1' is the estimate of 1'. (This assumes that you have used natural logarithms, that is, logarithms to base e, to transform the model.)
ELASTICITIES AND LOGARITHMIC MODELS
X
X
XY
loglog
loglog
loglog
21
1
1
2
2
21
XY
'' 2'1 XY
1'1 log
log'
,log'
XX
YYwhere
26
Here is a scatter diagram showing annual household expenditure on FDHO, food eaten at home, and EXP, total annual household expenditure, both measured in dollars, for 1995 for a sample of 869 households in the United States (Consumer Expenditure Survey data).
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FDHO
EXP
ELASTICITIES AND LOGARITHMIC MODELS
. reg FDHO EXP
Source | SS df MS Number of obs = 869---------+------------------------------ F( 1, 867) = 381.47 Model | 915843574 1 915843574 Prob > F = 0.0000Residual | 2.0815e+09 867 2400831.16 R-squared = 0.3055---------+------------------------------ Adj R-squared = 0.3047 Total | 2.9974e+09 868 3453184.55 Root MSE = 1549.5
------------------------------------------------------------------------------ FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- EXP | .0528427 .0027055 19.531 0.000 .0475325 .0581529 _cons | 1916.143 96.54591 19.847 0.000 1726.652 2105.634------------------------------------------------------------------------------
27
Here is a regression of FDHO on EXP. It is usual to relate types of consumer expenditure to total expenditure, rather than income, when using household data. Household income data tend to be relatively erratic.
ELASTICITIES AND LOGARITHMIC MODELS
28
The regression implies that, at the margin, 5 cents out of each dollar of expenditure is spent on food at home. Does this seem plausible? Probably, though possibly a little low.
ELASTICITIES AND LOGARITHMIC MODELS
. reg FDHO EXP
Source | SS df MS Number of obs = 869---------+------------------------------ F( 1, 867) = 381.47 Model | 915843574 1 915843574 Prob > F = 0.0000Residual | 2.0815e+09 867 2400831.16 R-squared = 0.3055---------+------------------------------ Adj R-squared = 0.3047 Total | 2.9974e+09 868 3453184.55 Root MSE = 1549.5
------------------------------------------------------------------------------ FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- EXP | .0528427 .0027055 19.531 0.000 .0475325 .0581529 _cons | 1916.143 96.54591 19.847 0.000 1726.652 2105.634------------------------------------------------------------------------------
29
It also suggests that $1,916 would be spent on food at home if total expenditure were zero. Obviously this is impossible. It might be possible to interpret it somehow as baseline expenditure, but we would need to take into account family size and composition.
ELASTICITIES AND LOGARITHMIC MODELS
. reg FDHO EXP
Source | SS df MS Number of obs = 869---------+------------------------------ F( 1, 867) = 381.47 Model | 915843574 1 915843574 Prob > F = 0.0000Residual | 2.0815e+09 867 2400831.16 R-squared = 0.3055---------+------------------------------ Adj R-squared = 0.3047 Total | 2.9974e+09 868 3453184.55 Root MSE = 1549.5
------------------------------------------------------------------------------ FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- EXP | .0528427 .0027055 19.531 0.000 .0475325 .0581529 _cons | 1916.143 96.54591 19.847 0.000 1726.652 2105.634------------------------------------------------------------------------------
30
Here is the regression line plotted on the scatter diagram
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ELASTICITIES AND LOGARITHMIC MODELS
FDHO
31
We will now fit a constant elasticity function using the same data. The scatter diagram shows the logarithm of FDHO plotted against the logarithm of EXP.
5.00
6.00
7.00
8.00
9.00
10.00
7.00 8.00 9.00 10.00 11.00 12.00 13.00
LGFDHO
LGEXP
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source | SS df MS Number of obs = 868---------+------------------------------ F( 1, 866) = 396.06 Model | 84.4161692 1 84.4161692 Prob > F = 0.0000Residual | 184.579612 866 .213140429 R-squared = 0.3138---------+------------------------------ Adj R-squared = 0.3130 Total | 268.995781 867 .310260416 Root MSE = .46167
------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- LGEXP | .4800417 .0241212 19.901 0.000 .4326988 .5273846 _cons | 3.166271 .244297 12.961 0.000 2.686787 3.645754------------------------------------------------------------------------------
32
Here is the result of regressing LGFDHO on LGEXP. The first two commands generate the logarithmic variables.
ELASTICITIES AND LOGARITHMIC MODELS
33
The estimate of the elasticity is 0.48. Does this seem plausible?
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source | SS df MS Number of obs = 868---------+------------------------------ F( 1, 866) = 396.06 Model | 84.4161692 1 84.4161692 Prob > F = 0.0000Residual | 184.579612 866 .213140429 R-squared = 0.3138---------+------------------------------ Adj R-squared = 0.3130 Total | 268.995781 867 .310260416 Root MSE = .46167
------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- LGEXP | .4800417 .0241212 19.901 0.000 .4326988 .5273846 _cons | 3.166271 .244297 12.961 0.000 2.686787 3.645754------------------------------------------------------------------------------
34
Yes, definitely. Food is a normal good, so its elasticity should be positive, but it is a basic necessity. Expenditure on it should grow less rapidly than expenditure generally, so its elasticity should be less than 1.
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source | SS df MS Number of obs = 868---------+------------------------------ F( 1, 866) = 396.06 Model | 84.4161692 1 84.4161692 Prob > F = 0.0000Residual | 184.579612 866 .213140429 R-squared = 0.3138---------+------------------------------ Adj R-squared = 0.3130 Total | 268.995781 867 .310260416 Root MSE = .46167
------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- LGEXP | .4800417 .0241212 19.901 0.000 .4326988 .5273846 _cons | 3.166271 .244297 12.961 0.000 2.686787 3.645754------------------------------------------------------------------------------
35
The intercept has no substantive meaning. To obtain an estimate of 1, we calculate e3.16, which is 23.8.
48.08.23ˆ48.017.3ˆ EXPOHFDLGEXPHODLGF
ELASTICITIES AND LOGARITHMIC MODELS
. g LGFDHO = ln(FDHO)
. g LGEXP = ln(EXP)
. reg LGFDHO LGEXP
Source | SS df MS Number of obs = 868---------+------------------------------ F( 1, 866) = 396.06 Model | 84.4161692 1 84.4161692 Prob > F = 0.0000Residual | 184.579612 866 .213140429 R-squared = 0.3138---------+------------------------------ Adj R-squared = 0.3130 Total | 268.995781 867 .310260416 Root MSE = .46167
------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval]---------+-------------------------------------------------------------------- LGEXP | .4800417 .0241212 19.901 0.000 .4326988 .5273846 _cons | 3.166271 .244297 12.961 0.000 2.686787 3.645754------------------------------------------------------------------------------
36
Here is the scatter diagram with the regression line plotted.
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7.00
8.00
9.00
10.00
7.00 8.00 9.00 10.00 11.00 12.00 13.00LGEXP
ELASTICITIES AND LOGARITHMIC MODELS
LGFDHO
37
Here is the regression line from the logarithmic regression plotted in the original scatter diagram, together with the linear regression line for comparison.
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ELASTICITIES AND LOGARITHMIC MODELS
FDHO
38
You can see that the logarithmic regression line gives a somewhat better fit, especially at low levels of expenditure.
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ELASTICITIES AND LOGARITHMIC MODELS
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However, the difference in the fit is not dramatic. The main reason for preferring the constant elasticity model is that it makes more sense theoretically. It also has a technical advantage that we will come to later on (when discussing heteroscedasticity).
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ELASTICITIES AND LOGARITHMIC MODELS
FDHO
Copyright Christopher Dougherty 2012.
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2012.11.03